Absolute Uniqueness of Phase Retrieval with Random Illumination

Random phase or amplitude illumination is proposed to remove at once all types of ambiguity, trivial or nontrivial, at once from phase retrieval. Almost sure irreducibility is proved for {\em any} complex-valued object of arbitrary sparsity. While this irreducibility result can be viewed as a probabilistic version of the classical result by Bruck, Sodin and Hayes, it provides a new perspective and an effective method for achieving absolute uniqueness in phase retrieval for {\em every} object, not just objects outside of a measure-zero set. In particular, almost sure absolute uniqueness is proved for complex-valued objects under a general two-point assumption. For objects of nonnegative real and imaginary parts, absolute uniqueness is proved to hold with probability exponentially close to unity as the object sparsity increases.